This paper considers spin - j representation of the Lie group SU(2) and its applications to contemporary physics from Observer's Mathematics point of view (see www.mathrelativity.com). This mathematics was introduced by authors based on denial of infinity idea. We proved here the following. Theorem. In Observers Mathematics the probability of spin - j transformation is a homomorphism (representation) of SU(2) to the set of matrix transformations of a linear space of polynomial functions is less than 1. Theorem. The probability of elementary fermions have half-integer spin in Observer's Mathematics is less than 1. The probability of elementary bosons have integer spin in Observer's Mathematics is less than 1.
Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a quantum circuit with at most
nc gates and each gate acting on a constant number of qubits, the sample complexity is bounded by
O~(nc+1). In particular, we explicitly construct a family of variational quantum circuits with
O(nc+1) elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most
nc elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and experiment.